Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 9.88% → 1.95%

Time: 11.6s

Precision: binary64

Cost: 13636

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.02e-98)
   (/ (/ 1.0 (hypot 1.0 z)) (* x (* (hypot 1.0 z) y)))
   (/ 1.0 (* y (+ x (* z (* x z)))))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-98) {
		tmp = (1.0 / hypot(1.0, z)) / (x * (hypot(1.0, z) * y));
	} else {
		tmp = 1.0 / (y * (x + (z * (x * z))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-98) {
		tmp = (1.0 / Math.hypot(1.0, z)) / (x * (Math.hypot(1.0, z) * y));
	} else {
		tmp = 1.0 / (y * (x + (z * (x * z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

↓

def code(x, y, z):
	tmp = 0
	if x <= -1.02e-98:
		tmp = (1.0 / math.hypot(1.0, z)) / (x * (math.hypot(1.0, z) * y))
	else:
		tmp = 1.0 / (y * (x + (z * (x * z))))
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.02e-98)
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) / Float64(x * Float64(hypot(1.0, z) * y)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(x + Float64(z * Float64(x * z)))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.02e-98)
		tmp = (1.0 / hypot(1.0, z)) / (x * (hypot(1.0, z) * y));
	else
		tmp = 1.0 / (y * (x + (z * (x * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[x, -1.02e-98], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(x + N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	9.88%
Target	8.1%
Herbie	1.95%

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e-98

   1. Initial program 4.21

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   2. Simplified 8.24

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right)}} \]
      Proof

      [Start] 4.21	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/l/ [=>] 4.75	\[ \color{blue}{\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]
      associate-*l* [=>] 8.24	\[ \frac{1}{\color{blue}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}} \]
      +-commutative [=>] 8.24	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z + 1\right)} \cdot x\right)} \]
      fma-def [=>] 8.24	\[ \frac{1}{y \cdot \left(\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)} \]

   3. Applied egg-rr 6.05

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\frac{\frac{1}{y}}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}} \]

   4. Simplified 2.91

      \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]
      Proof

      [Start] 6.05	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\frac{\frac{1}{y}}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}} \]
      associate-/l/ [=>] 5.99	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{1}{\left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot y}}}} \]
      associate-/r* [=>] 6.04	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}}} \]
      associate-/r* [=>] 5.82	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\frac{\color{blue}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{y}}} \]
      associate-/l* [<=] 5.84	\[ \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot y}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}} \]
      *-commutative [<=] 5.84	\[ \frac{1}{\frac{\color{blue}{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}} \]
      associate-/r* [<=] 6.07	\[ \frac{1}{\frac{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{1}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}} \]
      associate-/r/ [=>] 5.99	\[ \frac{1}{\color{blue}{\frac{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}{1} \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)}} \]
      /-rgt-identity [=>] 5.99	\[ \frac{1}{\color{blue}{\left(y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      fma-udef [=>] 5.99	\[ \frac{1}{\left(y \cdot \sqrt{\color{blue}{z \cdot z + 1}}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      +-commutative [<=] 5.99	\[ \frac{1}{\left(y \cdot \sqrt{\color{blue}{1 + z \cdot z}}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      hypot-1-def [=>] 5.99	\[ \frac{1}{\left(y \cdot \color{blue}{\mathsf{hypot}\left(1, z\right)}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      fma-udef [=>] 5.99	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \sqrt{\color{blue}{z \cdot z + 1}}\right)} \]
      +-commutative [<=] 5.99	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \sqrt{\color{blue}{1 + z \cdot z}}\right)} \]
      hypot-1-def [=>] 2.91	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \color{blue}{\mathsf{hypot}\left(1, z\right)}\right)} \]

   5. Applied egg-rr 2.11

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}} \]

   6. Simplified 0.68

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]
      Proof

      [Start] 2.11	\[ \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*r/ [=>] 0.43	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}} \]
      associate-*r/ [=>] 0.36	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot 1}{x}}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*l/ [<=] 0.36	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{x} \cdot 1}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*r/ [<=] 0.37	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{x} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)}} \]
      associate-/r* [<=] 2.33	\[ \color{blue}{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot x}} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/l/ [=>] 2.63	\[ \color{blue}{\frac{1}{\left(\mathsf{hypot}\left(1, z\right) \cdot x\right) \cdot y}} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*r* [<=] 1.74	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right) \cdot \left(x \cdot y\right)}} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)} \]
      *-commutative [<=] 1.74	\[ \frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \color{blue}{\left(y \cdot x\right)}} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*l/ [=>] 1.73	\[ \color{blue}{\frac{1 \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot x\right)}} \]
      *-lft-identity [=>] 1.73	\[ \frac{\color{blue}{\frac{1}{\mathsf{hypot}\left(1, z\right)}}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot x\right)} \]
      associate-*r* [=>] 0.68	\[ \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\color{blue}{\left(\mathsf{hypot}\left(1, z\right) \cdot y\right) \cdot x}} \]
      *-commutative [<=] 0.68	\[ \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\color{blue}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right)} \cdot x} \]
      *-commutative [=>] 0.68	\[ \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\color{blue}{x \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]

   if -1.02e-98 < x

   1. Initial program 17.78

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   2. Simplified 14.14

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right)}} \]
      Proof

      [Start] 17.78	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/l/ [=>] 17.99	\[ \color{blue}{\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]
      associate-*l* [=>] 14.14	\[ \frac{1}{\color{blue}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}} \]
      +-commutative [=>] 14.14	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z + 1\right)} \cdot x\right)} \]
      fma-def [=>] 14.14	\[ \frac{1}{y \cdot \left(\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)} \]

   3. Applied egg-rr 14.14

      \[\leadsto \frac{1}{y \cdot \color{blue}{\left(x \cdot \left(z \cdot z\right) + x\right)}} \]

   4. Taylor expanded in x around 0 14.14

      \[\leadsto \frac{1}{y \cdot \left(\color{blue}{{z}^{2} \cdot x} + x\right)} \]

   5. Simplified 3.74

      \[\leadsto \frac{1}{y \cdot \left(\color{blue}{z \cdot \left(z \cdot x\right)} + x\right)} \]
      Proof

      [Start] 14.14	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x + x\right)} \]
      unpow2 [=>] 14.14	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x + x\right)} \]
      associate-*l* [=>] 3.74	\[ \frac{1}{y \cdot \left(\color{blue}{z \cdot \left(z \cdot x\right)} + x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.95

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.06%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+15} \lor \neg \left(z \leq 4.4 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(x \cdot y\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2
Error	3.27%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;z \leq 1.12 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\left(\mathsf{hypot}\left(1, z\right) \cdot y\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \left(x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 3
Error	1.82%
Cost	1736

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 4
Error	6.82%
Cost	972

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 5
Error	2.87%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+108} \lor \neg \left(z \leq 110000000\right):\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 6
Error	3.77%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 7
Error	7.54%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 8
Error	4.12%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 9
Error	44.73%
Cost	320

\[\frac{1}{x \cdot y} \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))